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Let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space over $\mathbb{C}$ of all holomorphic functions $f:\mathbb{D}\rightarrow\mathbb{C}$. Define the following semi-norm:$$\left\Vert f\right\Vert \overset{\textrm{def}}{=}\sqrt{\lim_{r\uparrow1}\left(1-r\right)\int_{0}^{1}\left|f\left(re^{2\pi it}\right)\right|^{2}dt}$$ and let $\mathcal{H}\left(\mathbb{D}\right)$ denote the closure of the quotient of the subset of functions in $\mathcal{A}\left(\mathbb{D}\right)$ with finite semi-norm by the kernel of said semi-norm.

Now, writing $\mathbb{N}_{0}$ to denote the set of all non-negative integers, for any $V\subseteq\mathbb{N}_{0}$, write:$$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ I call such a function a “digital function” (its coefficients are $0$s and $1$s), a.k.a. the “set-series of $V$”. Then, define: $$\mathcal{D}\left(\mathbb{D}\right)\overset{\textrm{def}}{=}\left\{ \varsigma_{V}:V\subseteq\mathbb{N}_{0}\right\}$$ as the set of all digital functions.

With these definitions:

(1) Are there any functions in $\mathcal{A}\left(\mathbb{D}\right)$ which are not in $\mathcal{H}\left(\mathbb{D}\right)$?

(2) Is $\mathcal{H}\left(\mathbb{D}\right)$ dense in $\mathcal{A}\left(\mathbb{D}\right)$ (say, with respect to the norm topology on $\mathcal{H}\left(\mathbb{D}\right)$, or with respect to the topology of uniform convergence on compact subsets of $\mathbb{D}$)?

(3) The same thing as (1) and (2), but with $\mathcal{D}\left(\mathbb{D}\right)$ in place of $\mathcal{A}\left(\mathbb{D}\right)$. In particular, what can be said about $V$ if $\varsigma_{V}$ is not in $\mathcal{H}\left(\mathbb{D}\right)$?

One easy result is that sets of zero density fail to make an impact, since their associated set-series are in the kernel of the semi-norm. This makes me wonder: Consider the measure space of where the measurable sets are subsets of the non-negative integers with well-defined natural density. If we let $V$ be an arbitrary such set, does there then exist a set $W$ whose symmetric difference with $V$ has zero density such that $\varsigma_{W}$ is in $\mathcal{H}\left(\mathbb{D}\right)$ (or is the limit of a sequence of functions in $\mathcal{H}\left(\mathbb{D}\right)$)?

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    $\begingroup$ To me this is not research level, why don't you ask on MSE. Your $\|.\|$ is only a semi-norm for entire function $\|f\|=0$ so the natural map $\to H$ isn't injective. For your question (1) sure the $L^2$ norm of $e^{1/(z+1)}$ on $|z|=r$ isn't $O( (1-r)^{-1})$. $\endgroup$
    – reuns
    Commented Sep 3, 2019 at 23:49
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    $\begingroup$ I also don't understand why you get a well-defined seminorm, i.e. why should the limit in your definition be finite? $\endgroup$
    – Yemon Choi
    Commented Sep 4, 2019 at 0:21
  • $\begingroup$ @reuns: I know about semi-norms; I just goofed and failed to notice that this was only a semi-norm, not a norm. The counterexample didn't occur to me until just as I was going to sleep. xD $\endgroup$
    – MCS
    Commented Sep 4, 2019 at 20:29
  • $\begingroup$ @Yemon Choi: There exist a large family of functions for which the limit does exist. Those are the functions that I'm concerned with. For example, as a consequence of the Hardy-Littlewood Tauberian Theorem, all linear combinations of set-series of sets with well-defined natural density have finite semi-norm, as do all rational functions. $\endgroup$
    – MCS
    Commented Sep 4, 2019 at 20:32

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